Since fuzzy logic is, by definition, imprecise, it is a natural means of representing the imprecision of lattice parameters and bond angles. Fuzzy lattice parameters are created by collecting parameter values from the literature. From the minimum, maximum, and average of these parameter values, a fuzzy number is created that represents the imprecision in that parameter. Then, through the use of fuzzy arithmetic operators and recently developed fuzzy trigonometric functions, fuzzy atom locations within the unit cell and bond angles can be calculated. The benefit of using fuzzy logic in this manner is that it directly enables representation and calculation with the imprecision found in chemical compounds, which arises from measurement variability due to measurement error, structural defects, and thermal and vibrational characteristics.

There has been tremendous growth in the knowledge and understanding of chemistry and chemical compounds in this century. This knowledge has enabled researchers to develop advanced materials, such as polymers and composites, which have characteristics superior to naturally occurring materials. Yet, even with the most advanced equipment, we still cannot accurately model chemical properties, such as lattice parameters and bond angles, in simple, cubic-like structures, such as the chalcopyrite family of semiconducting compounds or the Y-Ba-Cu-O (YBCO) superconducting compound.

The literature is full of reports of lattice parameters and bond angles that are variable or imprecise for chemicals. This imprecision is a result of the analysis techniques used and the imprecision in the samples and measurement process. Research has shown that fuzzy logic is a unique tool for representing imprecision; this article extends fuzzy research into the area of molecular modeling.

Fuzzy set theory is a generalization of normal set theory and was introduced by Zadeh in 1965.1 (For a more in-depth look at fuzzy set theory, refer to References 2 and 3.)

In normal set theory, an object is either a member of a set or not (i.e., there are only two states), and the set is often referred to as a crisp set. Fuzzy sets, on the other hand, have degrees of membership to that set. Thus, it is possible for an object to have partial membership in a set. This forms the basis of fuzzy-set theory. Figure 1 shows a comparison of a crisp set and a fuzzy set. Notice that the number two is not a member of the crisp set 3, but it is a member of the fuzzy set . The vertical axis is called the degree of membership (a) and is normalized such that a [0, 1]. For example, the number 2.5 has a degree of membership of 0.5 in the fuzzy set in Figure 1.

Figure 2. An example of triangular fuzzy numbers, where = <2,3,5>.

Triangular fuzzy numbers (TFNs) are a subset of fuzzy sets with properties that make them well suited for modeling and design-type activities. Specifically, a TFN has a triangular shape represented by the triple <a, b, c> (Figure 2).

The left and right spreads of a TFN are a common measure of the TFN's variability or imprecision. In Figure 2, the left spread is 3  2 = 1; the right spread is 5  3 = 2.

One reason why TFNs are well suited to modeling and design is because their arithmetic operators and functions are developed, which allow fast operation on equations. These arithmetic operators include the following examples, where the following TFNs are given:

= <a1, a2, a3> and = <b1, b2, b3>

and the following values are used for illustration:

= <2, 4, 5> and = <1, 2, 3>

Addition

=
+
is defined as

= <c1, c2, c3> = <a1 + b1, a2 + b2, a3 + b3>

(1)

e.g., = <2 + 1, 4 + 2, 5 + 3> = <3, 6, 8>

Subtraction

=

is defined as

= <c1, c2, c3> = <a1  b3, a2  b2, a3  b1>

(2)

e.g., = <2  3, 4  2, 5  1> = <1, 2, 4>

Multiplication

=
x
is defined as

= <c1, c2, c3> = <a1 x b1, a2 x b2, a3 x b3>

(3)

e.g., = <2 x 1, 4 x 2, 5 x 3> = <2, 8, 15>

Division

= ÷ is defined as

= <c1, c2, c3> = <a1 ÷ b3, a2 ÷ b2, a3 ÷ b1>

(4)

e.g., = <2 ÷ 3, 4 ÷ 2, 5 ÷ 1> = <0.667, 2, 5>

Exponentiation

=
k (where k is the crisp scalar with, for example, a value of 0.5) is defined as

= <c1, c2, c3> = <a1k, a2k, a3k>

(5)

e.g., = <20.5, 40.5, 50.5> = <1.414, 2, 2.236>

In addition to these five arithmetic operators, recent research into fuzzy trigonometric functions has produced fuzzy versions of the cosine, sine, tangent, cotangent, secant, and cosecant functions (plus their respective inverses) that work with TFNs and the above arithmetic operators.4 Together, the arithmetic operators and trigonometric functions enable the modeling of chemical compounds and the calculation of fuzzy bond lengths and fuzzy bond angles.

Figure 3. A example of fuzzy lines where (top) point A is fixed, point B is fuzzy, and (bottom) point A is fuzzy, and point B is fuzzy. Also shown are values for the fuzzy endpoints.

Fuzzy molecular modeling (FMM) is the application of fuzzy logic to molecular modeling. Typically, to model a chemical's structure, lattice parameters and a coordinates table (such as the Wyckoff coordinates for a chemical compound under examination) are needed.5 Together, the lattice parameters and Wyckoff coordinates provide enough information to create a three-dimensional (3-D) structure of a compound's unit cell. However, the measurement of lattice parameters is subject to variability from one experiment to another. So, which value is correct in modeling the compound?

One solution is to take the average value, but that, in effect, creates a biased value with no retention of the variability of the data. Another approach is to create fuzzy lattice parameters and apply them to the Wyckoff coordinates. The fuzzy lattice parameters are created by searching the literature for experiments performed under the same conditions and then extracting the minimum, average, and maximum values for each lattice parameterthus creating TFNs. The end result is a fuzzy unit cell, with fuzzy bond lengths and fuzzy bond angles, that incorporates the variability found in literature. However, two interesting phenomena arise in creating fuzzy unit cellsthe concept of fuzzy lines and fuzzy vertices.

A fuzzy line is simply a line with imprecise endpoints. These lines are encountered nearly everyday in expressions like "about 10 m" or "close to 12 cm." If we represent the length of a line to be a TFN (e.g., = <8, 10, 12>Å), the endpoints must be determined to model a fuzzy unit cell. Two possible answers are shown in Figure 3.

a

b

Figure 4. The (a) 2-D and (b) 3-D fuzzy vertices.

In Figure 3, the first answer (top) to a fuzzy line has point A fixed (i.e., a crisp endpoint); point B is completely fuzzy. The second answer (bottom) has the two endpoints sharing the spread of the fuzzy line length. If each answer is checked via the distance formula, each set of endpoints will yield the starting line length. However, for this research, the second (bottom) answer provides a better representation of a fuzzy line in that each endpoint shares the total spread of the fuzzy line. This is clearly seen if the endpoints of two fuzzy unit cells are stacked together.

In evaluating fuzzy vertices, it is assumed that the fuzzy lines making up the fuzzy vertices intersect. For the two-dimensional (2-D) case, this creates a circular set of intersection points, and when the fuzzy vertice is extended to 3-D, there is a spherical space of intersection points. The 2-D and 3-D cases are shown in Figure 4. In each case, the intersection region is encapsulated by the union of the spreads of the intersecting fuzzy lines.

The chalcopyrite family of compounds comprise semiconductors that possess beneficial electro-optical properties. A chalcopyrite compound, denoted as ABC2, is composed of the cations A and B and the anion C. The unit cell of a chalcopyrite-type compound has 13 A atoms, 10 B atoms, and 8 C atoms. As the chalcopyrite family features more than 30 compounds, the compound CuInSe2 was selected for modeling purposes.

Using values extracted from literature,612 the fuzzy lattice parameters are

ã = <5.773, 5.781, 5.785> Å

= <5.773, 5.781, 5.785> Å

= <11.550, 11.602, 11.642> Å

and the anion displacement parameter is

= <0.220, 0.230, 0.249>

Figure 5. A QuickTime movie of the chalcopyrite unit cell, ABC2. The red atoms represent the A-type atoms, the green atoms are the B-type atoms, and the blue atoms are the C-type atoms. Hence, for CuInSe2, the red atoms are copper, the green atoms are indium, and the blue atoms are selenium. (~850 k)

Using the fuzzy lattice parameters and the Wyckoff coordinates, fuzzy unit cells of CuInSe2 (Figure 5, an animation) and other compounds can be calculated, as depicted by a Java applet found on the supplemental page "Fuzzy" Applets.

The "Fuzzy" Applets page also contains a Java applet for calculating the fuzzy density of a chalcopyrite compound. While most of the properties of chemical compounds have indirect relationships with the lattice parameters, density has a direct relationship. For example, if considering CuInSe2, the fuzzy lattice parameters can be used to calculate a fuzzy representation of the volume of the unit cell. From a standard periodic table of elements, the mass of copper, indium, and selenium is used to find the total mass of the fuzzy unit cell, which is 2.234 x 1021 grams. As density is the ratio of mass over volume, the fuzzy density for CuInSe2 is <5.733, 5.761, 5.803> g/cm3. Thus, the density may be as low as 5.733 g/cm3 or as high as 5.803 g/cm3, but we would expect it to be nominally 5.761 g/cm3.

The superconducting compound YBCO exists in many forms, as the oxygen content can vary. The research discussed here focused on the orthorhombic structure of the high-temperature superconducting compound YBa2Cu3O7. Proceeding as for the chalcopyrite unit cell, a set of values extracted from the literature1419 was used to create the fuzzy lattice parameters. They are

ã = <3.821, 3.830, 3.856> Å

= <3.870, 3.882, 3.890> Å

= <11.666, 11.687, 11.708> Å

Using the fuzzy lattice parameters and the Wyckoff coordinates for the Pmmm space group,20 a fuzzy unit cell for the YBa2Cu3O7 compound was constructed (Figure 6, an animation).

By incorporating the imprecision found in the literature to the modeling of chemical unit cells using TFNs, we acknowledge the measurement error, structural defects, and thermal and vibrational effects that affectand can possibly alterthe characteristics associated with a unit cell's structure. Additionally, by using fuzzy lattice parameters, it is possible to express other characteristics of the unit cell in imprecise terms, thereby providing researchers with possible ranges for characteristics of interest. This has the added benefit of providing variable information that can help simplify a decision of selecting a specific compound out of a selection of many.

It is believed that this research is a necessary first step to representing the imprecision found in unit cells and will, hopefully, lead to a better understanding of chemical properties.